Curves of genus 2 with split Jacobian

We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X X is a curve of genus 2 2 , and f : X → E f:X → E a map from X X to an elliptic curve, then X X has split jacobian. It is not true that a complement to E E in the jacobian of X X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve E ′ E’ and algebraic f : X → E ′ f:X → E’ , and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 0 and 3 3 .